Does weak convergence with uniformly bounded densities imply absolute continuity of the limit?

Suppose $(f_n)$ is a sequence of probability density functions on $\mathbb{R}$ such that \begin{align*} f_n &\leq M \text{ for all }n \\ f_n(x) &= 0 \text{ for all } |x| > 1 \end{align*} Suppose also that there is a Borel probability measure $\mu$ on $\mathbb{R}$ such that $$\lim_{n \rightarrow \infty} \int gf_n dx = \int g d\mu \quad \forall g \in C_b(\mathbb{R}),$$ where $C_b(\mathbb{R})$ is the set of all bounded, continuous, complex-valued functions on $\mathbb{R}$. (In other words, suppose the measures $\mu_n$ defined by $d\mu_n = f_n dx$ converge to $\mu$ weakly.)

Question: Is $\mu$ absolutely continuous?

Yes, it is.

Step 1 We have $\mu(A)=0$ for any $A \in \mathcal{B}(\mathbb{R})$ such that $$d(A,[-1,1]) := \inf_{\substack{x \in A \\ y \in [-1,1]}} |x-y|>0.$$ Indeed: Pick a cut-off function $\chi_k \in C_b$ satisfying $\chi(x)=0$ for $|x| \leq 1$, $\chi(x)=1$ for $|x| > 1+1/k$. Since $f_n(x)=0$ for $x \in \mathbb{R} \backslash [-1,1]$ the weak convergence implies

$$\int \chi_k(x) \, d\mu(x) = \lim_{n \to \infty} \int \chi_k(x) \cdot f_n(x) \, dx =0$$

Hence,

$$\mu(A) = \int 1_A \, d\mu \leq \int \chi_k(x) \, d\mu(x) =0$$

for any $K$ such that $d(A,[-1,1]) > 1/k$.

Step 2 By the Radon-Nikodym Theorem, $\mu$ is absolutely continous with respect to the Lebesgue measure $\lambda$, if and only if, $\lambda(A)=0$ implies $\mu(A)=0$ for each $A \in \mathcal{B}(\mathbb{R})$. Since the first part of the proof implies in particular $\mu(\mathbb{R} \backslash [-2,2])=0$, it suffices to show $\lambda(A)=0$ implies $\mu(A)=0$ for $A \subseteq [-2,2]$.

Let $A \subseteq [-2,2]$ be open. We define

$$A_k := \{x \in \mathbb{R}; |x| \leq 2; d(x,A^c) \geq 1/k\}, \qquad k \in \mathbb{N}$$

Obviously, $A_k$ is compact and $A_k \uparrow A$. It is not difficult to show that there exists a sequence $(g_k)_{k \in \mathbb{N}} \subseteq C_C$ such that $1_{A_k} \leq g_k \leq 1_A$ and $g_k \uparrow 1_A$. Therefore, we find by the monotone convergence theorem and the weak convergence

$$\mu(A) = \sup_{k \in \mathbb{N}} \int g_k \, d\mu = \sup_{k \in \mathbb{N}} \limsup_{n \to \infty} \int g_k(x) \cdot f_n(x) \, dx$$

Since $0 \leq f_n \leq M$ and $g_k \uparrow 1_A$, we find

$$\mu(A) \leq \limsup_{n \to \infty} \int 1_A(x) \cdot f_n(x) \, dx \leq M \cdot \lambda(A) \tag{1}$$

Recall that a finite measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is outer regular, i.e.

$$\lambda(B) = \inf\{\lambda(A); A \supseteq B, A \, \text{open}\} \tag{2}$$

By $(1)$ we find that for any $B \in \mathcal{B}(\mathbb{R})$ and $A \supseteq B$ open,

$$\mu(B) \leq \mu(A) \leq M \cdot \lambda(A)$$

Consequently, $(2)$ entails $\mu(B)=0$ for any set $B$ satisfying $\lambda(B)=0$.

Remark Note that the assumption on the uniform boundedness is crucial, see for instance this question.

• Hello ! Nice answer. Would you have a reference for this proof, please (I mean a textbook or a monograph where such a proof and similar results are shown)? – MoebiusCorzer May 14 '17 at 11:58
• @MoebiusCorzer Sorry, I don't have a reference for the result or the proof. Billingsley ("Convergence of probability measures") discusses weak convergence quite thoroughly, but I'm not sure whether this result is in there. – saz May 14 '17 at 12:55
• Thanks for your prompt answer. I have checked Billingsley's book but haven't found a proof. It is not a problem, I will adapt it to my problem. – MoebiusCorzer May 14 '17 at 12:56

One may also prove this via functional analysis. Indeed, for $g \in C^{\infty}([-1,1])$, let us define $$Q(g):= \int g d\mu.$$ By the given condition and Cauchy-Schwarz, we have that $$|Q(g)| = \lim_{n \to \infty} \big| \langle f_n,g \rangle_{L^2} \big|\leq \limsup_{n \to \infty} \|f_n\|_{L^2} \|g\|_{L^2} \leq 2^{1/2}M \|g\|_{L^2},$$ so that $Q$ can be extended to a bounded linear operator from $L^2[-1,1] \to \Bbb R$. By the Riesz representation theorem, we know any bounded linear functional on $L^2$ is given by $\langle f, \cdot \rangle$ for some $f$. Thus there exists $f \in L^2[-1,1]$ such that for all $g \in L^2[-1,1]$ we have $$Q(g) = \int fg$$which then implies that $\mu$ is absolutely continuous with density $f$.

• I really liked your approach. Could you please help me understand the last sentence though? Why $\mu$ is absolutely continuous with density $f$? – kaithkolesidou Feb 20 at 11:23
• @kaithkolesidou the point is, if $\int gf \; dx = \int g \;d\mu$ for all continuous $g$, then one necessarily has that the measure $f\;dx$ must agree with $\mu$, since measures are determined by their integrals against continuous test functions. – Shalop Feb 20 at 15:16
• yes I can see that. But this is equivalent to absolutely continuity because there exists a measurable $f$ such that $\mu=fdx$? I thought only the opposite direction is true – kaithkolesidou Feb 20 at 15:22
• @kaithkolesidou I think your understanding of absolute continuity is backwards. – Shalop Feb 20 at 21:16
• As I see it, $\mu$ is $\sigma$-finite on $[-1,1]$ and thus by $d\mu=fdx$ we can deduce by Rdon-Nikodym theorem the absolutely continuity of $\mu$. Please correct me if I 'm wrong because this is what I'm missing... However thanks a lot already for your time! – kaithkolesidou Feb 21 at 10:30